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On Counting Lattice Points and Chvátal-Gomory Cutting Planes [chapter]

Andrea Lodi, Gilles Pesant, Louis-Martin Rousseau
2011 Lecture Notes in Computer Science  
The paper investigates the relationship between counting the lattice points belonging to an hyperplane and the separation of Chvátal-Gomory cutting planes.  ...  In particular, we show that counting can be exploited in two ways: (i) to strengthen the cuts separated, e.g., by the classical procedure of Gomory, and (ii) to heuristically evaluate the effectiveness  ...  The authors are grateful to Matteo Fischetti and Juliane Dunkel for interesting discussions on the topic.  ... 
doi:10.1007/978-3-642-21311-3_13 fatcat:niqgeukq6fg3jjt7jlsmxvjaoe

On the Rank of Cutting-Plane Proof Systems [chapter]

Sebastian Pokutta, Andreas S. Schulz
2010 Lecture Notes in Computer Science  
This was later used to obtain a lower bound of (1 + )n on the Gomory-Chvátal rank of arbitrary polytopes P ⊆ [0, 1] n , showing that in contrast to most other cutting-plane procedures, the Gomory-Chvátal  ...  This leads to a new class of proof systems that includes many well-known methods, such as Gomory-Chvátal cuts, lift-and-project cuts, Sherali-Adams cuts, or split cuts.  ...  These include Gomory-Chvátal cuts Chvátal [1973] , Gomory [1958] , the lift-and-project cuts of Balas, Ceria and Cornuéjols Balas et al. [1993] , Sherali-Adams cuts Sherali and Adams [1990] , the matrix  ... 
doi:10.1007/978-3-642-13036-6_34 fatcat:626wcllt6fbczbjdj5zv557zo4

Scanning integer points with lex-inequalities: A finite cutting plane algorithm for integer programming with linear objective [article]

Michele Conforti, Marianna De Santis, Marco Di Summa, Francesco Rinaldi
2020 arXiv   pre-print
The family of lex-cuts contains the Chvatal-Gomory cuts, but does not contain and is not contained in the family of split cuts.  ...  We consider the integer points in a unimodular cone K ordered by a lexicographic rule defined by a lattice basis.  ...  We are also grateful to Akshay Gupte for his constructive comments and his pointers to the existing literature.  ... 
arXiv:1811.02345v2 fatcat:hmfu2hilizcnvce4cip3ijv4zi

Scanning integer points with lex-inequalities: a finite cutting plane algorithm for integer programming with linear objective

Michele Conforti, Marianna De Santis, Marco Di Summa, Francesco Rinaldi
2020 4OR  
The family of lex-inequalities contains the ChvátalGomory cuts, but does not contain and is not contained in the family of split cuts.  ...  AbstractWe consider the integer points in a unimodular cone K ordered by a lexicographic rule defined by a lattice basis.  ...  Acknowledgements The lex-cuts defined in a previous version of this manuscript were weaker. Giacomo Zambelli suggested to derive stronger lex-cuts via the characterization of the polyhedron Q(x).  ... 
doi:10.1007/s10288-020-00459-6 fatcat:s57tueauundszdnm7qfarj74su

Helly systems and certificates in optimization [article]

Amitabh Basu, Tongtong Chen, Michele Conforti, Hongyi Jiang
2022 arXiv   pre-print
Inspired by branch-and-bound and cutting plane proofs in mixed-integer optimization and proof complexity, we develop a general approach via Hoffman's Helly systems.  ...  The second part of the paper establishes lower and upper bounds on the sizes of these certificates in various different settings.  ...  system (e.g., Chvátal-Gomory cutting planes or disjunctive cuts).  ... 
arXiv:2111.05225v3 fatcat:elfvauedlrd47ogpg22vr4tdm4

On Polytopes with Linear Rank with respect to Generalizations of the Split Closure [article]

Sanjeeb Dash, Yatharth Dubey
2021 arXiv   pre-print
n) with respect to t-dimensional lattice cuts.  ...  In this paper we study the rank of polytopes contained in the 0-1 cube with respect to t-branch split cuts and t-dimensional lattice cuts for a fixed positive integer t.  ...  These polytopes were studied in the context of other families of cutting planes in [BOGH + 03] , namely Gomory-Chvátal (GC) cuts, and the matrix cuts based on the 0 , , and + [LS91] operators.  ... 
arXiv:2110.04344v2 fatcat:2yp6x7vxv5gpva4po6fn2e7fy4

Small Chvatal rank [article]

Tristram Bogart, Annie Raymond, Rekha R. Thomas
2009 arXiv   pre-print
We propose a variant of the Chvatal-Gomory procedure that will produce a sufficient set of facet normals for the integer hulls of all polyhedra xx : Ax <= b as b varies.  ...  Lower bounds for SCR are derived both in general and for polytopes in the unit cube.  ...  We thank Sasha Barvinok, Ravi Kannan and Les Trotter for helpful inputs to this paper.  ... 
arXiv:0705.1027v3 fatcat:qyh62smcnzbzvjcoebj6kyo6xu

Mixed-integer linear representability, disjunctions, and Chvatal functions --- modeling implications [article]

Amitabh Basu, Kipp Martin, Christopher Thomas Ryan, Guanyi Wang
2017 arXiv   pre-print
This allows us to answer a long-standing open question due to Ryan (1991) on designing an elimination scheme to represent finitely-generated integral monoids as a system of Chvatal inequalities without  ...  Unlike the case for linear inequalities, allowing for integer variables in Chvatal inequalities and projection does not enhance modeling power.  ...  Acknowledgments We thank the anonymous reviewers who commented on a preliminary version of this manuscript that appears in the 2017 proceedings of the IPCO conference (referenced here as [5] ).  ... 
arXiv:1711.07028v1 fatcat:nxew3qvhkzhm5jzin4vdqjlm5q

Mixed-integer bilevel representability [article]

Amitabh Basu, Christopher Thomas Ryan, Sriram Sankaranarayanan
2018 arXiv   pre-print
Conversely, any finite union of polyhedra can be represented using any one of these three paradigms.  ...  We then prove that the feasible region of bilevel problems with integer constraints exclusively in the upper level is a finite union of sets representable by mixed-integer programs and vice versa.  ...  Acknowledgments The first and third authors are supported by NSF grant CMMI1452820 and ONR grant N000141812096.  ... 
arXiv:1808.03865v3 fatcat:qmvsgqinorhmrj3l7x6dlvijmy

Enhanced mixed integer programming techniques and routing problems

Andrea Tramontani
2010 4OR  
plane capability, and in particular Gomory mixed-integer cuts.  ...  ) and a better lower bound at the root node due to the use of Chvátal-Gomory cuts (2 units improvement on a absolute gap of 22 units).  ...  branching rules, primal-heuristic algorithms and the use of general-purpose Chvátal-Gomory cuts.  ... 
doi:10.1007/s10288-010-0140-x fatcat:x7vxwnkff5csbbuq7nqfnjbfpi

Theoretical challenges towards cutting-plane selection [article]

Santanu S. Dey, Marco Molinaro
2018 arXiv   pre-print
a portfolio of cutting-planes to be added to the LP relaxation at a given node of the branch-and-bound tree.  ...  In this paper we review the different classes of cutting-planes available, known theoretical results about their relative strength, important issues pertaining to cut selection, and discuss some possible  ...  Acknowledgements We would like to thank Tobias Achterberg, Domenico Salvagnin and Roland Wunderling for their help with preparing this manuscript.  ... 
arXiv:1805.02782v1 fatcat:xc62crye3vbkvdwxgf2b2a5ycq

Cutting planes from extended LP formulations

Merve Bodur, Sanjeeb Dash, Oktay Günlük
2016 Mathematical programming  
We then extend this idea to general mixed-integer sets and construct the best extended LP formulation for such sets with respect to lattice-free cuts.  ...  of adding cutting planes in the extended space.  ...  Acknowledgements We would like to thank the anonymous referees for their careful reading of the paper and constructive comments. We also thank Jim Luedtke for useful discussions on this topic.  ... 
doi:10.1007/s10107-016-1005-7 fatcat:h4thok7xcngozcl6u2ljr2x6xi

Computing convex hulls and counting integer points with polymake

Benjamin Assarf, Ewgenij Gawrilow, Katrin Herr, Michael Joswig, Benjamin Lorenz, Andreas Paffenholz, Thomas Rehn
2016 Mathematical Programming Computation  
The main purpose of this paper is to report on the state of the art of computing integer hulls and their facets as well as counting lattice points in convex polytopes.  ...  Using the polymake system we explore various algorithms and implementations. Our experience in this area is summarized in ten "rules of thumb".  ...  ., for normaliz' handling of non-symmetric cut polytopes.  ... 
doi:10.1007/s12532-016-0104-z fatcat:qk3iih53zfc3fdscu54lh6wwey

A primal all-integer algorithm based on irreducible solutions

Utz-Uwe Haus, Matthias K�ppe, Robert Weismantel
2003 Mathematical programming  
The algorithm iteratively substitutes one column in a tableau by other columns that correspond to irreducible solutions of certain linear diophantine inequalities.  ...  While the optimal solution to the relaxation is not integral, one continues adding cutting planes to the problem formulation and reoptimizes.  ...  We prove that it is finite and illustrate the advantage of our algorithm on two examples when compared to a pure LP-based branch-and-bound procedure and a pure cutting-plane algorithm.  ... 
doi:10.1007/s10107-003-0384-8 fatcat:xffb4xz55vfo7kx3uskwxu4hra

Computing convex hulls and counting integer points with polymake [article]

Benjamin Assarf, Ewgenij Gawrilow, Katrin Herr, Michael Joswig, Benjamin Lorenz, Andreas Paffenholz, Thomas Rehn
2015 arXiv   pre-print
The main purpose of this paper is to report on the state of the art of computing integer hulls and their facets as well as counting lattice points in convex polytopes.  ...  Using the polymake system we explore various algorithms and implementations. Our experience in this area is summarized in ten "rules of thumb".  ...  Moreover, we are very grateful to the developers of cdd, lrs, normaliz and ppl as they gave us various kind of valuable feedback. The comments by David Avis and Winfried  ... 
arXiv:1408.4653v2 fatcat:vzc2o22kj5b4fimjfqrlk5opiq
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